The smallest gap between primes

A prime gap is the difference between two successive prime numbers. Two is the smallest possible gap between primes. A twin prime is a prime that has a prime gap of two. The twin prime conjecture states that there are infinitely many twin primes. This conjecture has been one of the great open problems in number theory for many years. In May 2013, the popular Yitang Zhang's paper was accepted by the journal Annals of Mathematics where it was announced that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N. A few months later, James Maynard gave a different proof of Yitang Zhang's theorem and showed that there are infinitely many prime gaps with size of at most 600. A collaborative effort in the Polymath Project, led by Terence Tao, reduced to the lower bound 246 just using Zhang and Maynard results as the main theoretical background. In this note, using arithmetic operations, we prove that the twin prime conjecture is true. Indeed, this is a trivial and short note very easy to check and understand which is a breakthrough result at the same time.